I want to plot the following vector field in 2D, using the built in function VectorPlot: Because the variation in modulus is typically several orders of magnitude, Mathematica's arrows are quite large near the source point, the point , and are very short at the edge of the plot.

The way to solve this issue is to have the arrow lengths scaled logarithmically.

How can one do that in Mathematica? I came across the VectorScale option, and saw it accepts 3 arguments:

{unitlen,aratio,sfun}

As stated in the Documentation,

sfun is a scaling function that determines the width of vector boxes.

I think that the argument that is in interest for my purpose is sfun. But I need some help with writing down a command to be given there. Any help about it would be very appreciated.